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128      XIII   INTEGRATION

(5)    there exists an integrable function g g: 0 such that, for all z e E, we
have \D2f(x, z)\ 5* 9i(x) almost everywhere in X.

Then h is differentiable at every point ofE, and we have
(13.8.6.1)                       h'(z) = f D2/(x, z) dtfx)

("differentiation under the.integral sign").

(iii)   Now suppose that E is an open set in C and that f is a mapping of
X x E into C, satisfying the following conditions:

(6)   for almost all x e X, the function zi>f(x, z) is analytic in E;

(7)   for all z e E, the function x H-/(JC, z) is integrable ;

(8)    there exists an integrable function g^O such that for all z e E we
have \f(x9 z)| <j! g(x) almost everywhere in X.

Then h is analytic in E, and for each integer k > 0 and each point z e E,
the function x\~* D*/(^, z) fa integrable, and we have

(i)   By (3.13.14) it is enough to show that, for every sequence (zn) tending
to z0 in E, we have lim h(zn) = A(z0); and this follows from the hypotheses

B-^OO

and from (13.8.4).

(ii)   Again by virtue of (3.13.14), it is enough to show that, for each
sequence (O of real numbers =0 and tending to 0, the sequence

has as limit the right-hand side of (13.8.6.1). But it follows from the hypotheses
and from (8.5.4) that

for all n} so that the result again follows from (13.8.4).

(iii)   Let y : t\-+a + relt (0 <; t g 2n) be a circuit contained in E. For
each z in the interior of the disc |z  a\ < r, we have

(13.8.6.2)                     f(x, z) =  I /V*'M

V                 /                                      / V^J **/         /^__-    I         f

~~" Zen h(z) = \f(x, z) dfi(x) is continuous at ZQ .